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Theorem signslema 29456
Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)
Hypotheses
Ref Expression
signslema.1  |-  ( ph  ->  E  e.  NN0 )
signslema.2  |-  ( ph  ->  F  e.  NN0 )
signslema.3  |-  ( ph  ->  G  e.  NN0 )
signslema.4  |-  ( ph  ->  H  e.  NN0 )
signslema.5  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
signslema.6  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
Assertion
Ref Expression
signslema  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )

Proof of Theorem signslema
StepHypRef Expression
1 signslema.5 . . . . . 6  |-  ( ph  ->  ( E  <  G  /\  -.  2  ||  ( G  -  E )
) )
21simpld 465 . . . . 5  |-  ( ph  ->  E  <  G )
32adantr 471 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  E  <  G )
4 signslema.4 . . . . . . . . . 10  |-  ( ph  ->  H  e.  NN0 )
54nn0cnd 10916 . . . . . . . . 9  |-  ( ph  ->  H  e.  CC )
6 signslema.2 . . . . . . . . . 10  |-  ( ph  ->  F  e.  NN0 )
76nn0cnd 10916 . . . . . . . . 9  |-  ( ph  ->  F  e.  CC )
85, 7subcld 9972 . . . . . . . 8  |-  ( ph  ->  ( H  -  F
)  e.  CC )
9 signslema.3 . . . . . . . . . 10  |-  ( ph  ->  G  e.  NN0 )
109nn0cnd 10916 . . . . . . . . 9  |-  ( ph  ->  G  e.  CC )
11 signslema.1 . . . . . . . . . 10  |-  ( ph  ->  E  e.  NN0 )
1211nn0cnd 10916 . . . . . . . . 9  |-  ( ph  ->  E  e.  CC )
1310, 12subcld 9972 . . . . . . . 8  |-  ( ph  ->  ( G  -  E
)  e.  CC )
148, 13subeq0ad 9982 . . . . . . 7  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  <-> 
( H  -  F
)  =  ( G  -  E ) ) )
1514biimpa 491 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( H  -  F )  =  ( G  -  E ) )
1615breq2d 4385 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
0  <  ( H  -  F )  <->  0  <  ( G  -  E ) ) )
176nn0red 10915 . . . . . . 7  |-  ( ph  ->  F  e.  RR )
184nn0red 10915 . . . . . . 7  |-  ( ph  ->  H  e.  RR )
1917, 18posdifd 10188 . . . . . 6  |-  ( ph  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2019adantr 471 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
2111nn0red 10915 . . . . . . 7  |-  ( ph  ->  E  e.  RR )
229nn0red 10915 . . . . . . 7  |-  ( ph  ->  G  e.  RR )
2321, 22posdifd 10188 . . . . . 6  |-  ( ph  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2423adantr 471 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
2516, 20, 243bitr4rd 294 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  ( E  <  G  <->  F  <  H ) )
263, 25mpbid 215 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  F  <  H )
27 0red 9630 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  e.  RR )
2822, 21resubcld 10035 . . . . . 6  |-  ( ph  ->  ( G  -  E
)  e.  RR )
2928adantr 471 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  e.  RR )
3018, 17resubcld 10035 . . . . . 6  |-  ( ph  ->  ( H  -  F
)  e.  RR )
3130adantr 471 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( H  -  F )  e.  RR )
322adantr 471 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  E  <  G )
3323adantr 471 . . . . . 6  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( E  <  G  <->  0  <  ( G  -  E ) ) )
3432, 33mpbid 215 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( G  -  E
) )
35 2pos 10689 . . . . . . 7  |-  0  <  2
36 breq2 4377 . . . . . . 7  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  (
0  <  ( ( H  -  F )  -  ( G  -  E ) )  <->  0  <  2 ) )
3735, 36mpbiri 241 . . . . . 6  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  =  2  ->  0  <  ( ( H  -  F )  -  ( G  -  E )
) )
3828, 30posdifd 10188 . . . . . . 7  |-  ( ph  ->  ( ( G  -  E )  <  ( H  -  F )  <->  0  <  ( ( H  -  F )  -  ( G  -  E
) ) ) )
3938biimpar 492 . . . . . 6  |-  ( (
ph  /\  0  <  ( ( H  -  F
)  -  ( G  -  E ) ) )  ->  ( G  -  E )  <  ( H  -  F )
)
4037, 39sylan2 481 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( G  -  E )  <  ( H  -  F
) )
4127, 29, 31, 34, 40lttrd 9782 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  0  <  ( H  -  F
) )
4219adantr 471 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  ( F  <  H  <->  0  <  ( H  -  F ) ) )
4341, 42mpbird 240 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  F  <  H )
445, 10, 7, 12sub4d 10021 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  =  ( ( H  -  F )  -  ( G  -  E ) ) )
45 signslema.6 . . . . 5  |-  ( ph  ->  ( ( H  -  G )  -  ( F  -  E )
)  e.  { 0 ,  2 } )
4644, 45eqeltrrd 2530 . . . 4  |-  ( ph  ->  ( ( H  -  F )  -  ( G  -  E )
)  e.  { 0 ,  2 } )
47 ovex 6303 . . . . 5  |-  ( ( H  -  F )  -  ( G  -  E ) )  e. 
_V
4847elpr 3953 . . . 4  |-  ( ( ( H  -  F
)  -  ( G  -  E ) )  e.  { 0 ,  2 }  <->  ( (
( H  -  F
)  -  ( G  -  E ) )  =  0  \/  (
( H  -  F
)  -  ( G  -  E ) )  =  2 ) )
4946, 48sylib 201 . . 3  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  0  \/  ( ( H  -  F )  -  ( G  -  E
) )  =  2 ) )
5026, 43, 49mpjaodan 800 . 2  |-  ( ph  ->  F  <  H )
511simprd 469 . . . . 5  |-  ( ph  ->  -.  2  ||  ( G  -  E )
)
5251adantr 471 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( G  -  E ) )
5315breq2d 4385 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  (
2  ||  ( H  -  F )  <->  2  ||  ( G  -  E
) ) )
5452, 53mtbird 307 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  0 )  ->  -.  2  ||  ( H  -  F ) )
55 2z 10958 . . . . . . 7  |-  2  e.  ZZ
569nn0zd 11027 . . . . . . . 8  |-  ( ph  ->  G  e.  ZZ )
5711nn0zd 11027 . . . . . . . 8  |-  ( ph  ->  E  e.  ZZ )
5856, 57zsubcld 11034 . . . . . . 7  |-  ( ph  ->  ( G  -  E
)  e.  ZZ )
59 dvdsaddr 14354 . . . . . . 7  |-  ( ( 2  e.  ZZ  /\  ( G  -  E
)  e.  ZZ )  ->  ( 2  ||  ( G  -  E
)  <->  2  ||  (
( G  -  E
)  +  2 ) ) )
6055, 58, 59sylancr 674 . . . . . 6  |-  ( ph  ->  ( 2  ||  ( G  -  E )  <->  2 
||  ( ( G  -  E )  +  2 ) ) )
6151, 60mtbid 306 . . . . 5  |-  ( ph  ->  -.  2  ||  (
( G  -  E
)  +  2 ) )
6261adantr 471 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( ( G  -  E )  +  2 ) )
63 2cnd 10670 . . . . . . 7  |-  ( ph  ->  2  e.  CC )
648, 13, 63subaddd 9990 . . . . . 6  |-  ( ph  ->  ( ( ( H  -  F )  -  ( G  -  E
) )  =  2  <-> 
( ( G  -  E )  +  2 )  =  ( H  -  F ) ) )
6564biimpa 491 . . . . 5  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
( G  -  E
)  +  2 )  =  ( H  -  F ) )
6665breq2d 4385 . . . 4  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  (
2  ||  ( ( G  -  E )  +  2 )  <->  2  ||  ( H  -  F
) ) )
6762, 66mtbid 306 . . 3  |-  ( (
ph  /\  ( ( H  -  F )  -  ( G  -  E ) )  =  2 )  ->  -.  2  ||  ( H  -  F ) )
6854, 67, 49mpjaodan 800 . 2  |-  ( ph  ->  -.  2  ||  ( H  -  F )
)
6950, 68jca 539 1  |-  ( ph  ->  ( F  <  H  /\  -.  2  ||  ( H  -  F )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1447    e. wcel 1890   {cpr 3937   class class class wbr 4373  (class class class)co 6275   RRcr 9524   0cc0 9525    + caddc 9528    < clt 9661    - cmin 9846   2c2 10647   NN0cn0 10858   ZZcz 10926    || cdvds 14315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570  ax-resscn 9582  ax-1cn 9583  ax-icn 9584  ax-addcl 9585  ax-addrcl 9586  ax-mulcl 9587  ax-mulrcl 9588  ax-mulcom 9589  ax-addass 9590  ax-mulass 9591  ax-distr 9592  ax-i2m1 9593  ax-1ne0 9594  ax-1rid 9595  ax-rnegex 9596  ax-rrecex 9597  ax-cnre 9598  ax-pre-lttri 9599  ax-pre-lttrn 9600  ax-pre-ltadd 9601  ax-pre-mulgt0 9602
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-pss 3387  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-tp 3940  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-tr 4469  df-eprel 4722  df-id 4726  df-po 4732  df-so 4733  df-fr 4770  df-we 4772  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-pred 5358  df-ord 5404  df-on 5405  df-lim 5406  df-suc 5407  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6680  df-wrecs 7014  df-recs 7076  df-rdg 7114  df-er 7349  df-en 7556  df-dom 7557  df-sdom 7558  df-pnf 9663  df-mnf 9664  df-xr 9665  df-ltxr 9666  df-le 9667  df-sub 9848  df-neg 9849  df-nn 10598  df-2 10656  df-n0 10859  df-z 10927  df-dvds 14316
This theorem is referenced by: (None)
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